It is the end of my first year of teaching, and I am in a reflective mood. The art of "reflection" was one heavily mocked in my professional college. It seemed as though every assignment in the College of Education involved some kind of reflection. Students of other colleges dismissed the idea as elementary. Do something useful, then reflect on it, then reflect on that reflection, etc. The process began to resemble an infinite sequence. It wasn't until the reflections were no longer forced, that I found value in the process.
My year began in chaos. I was hired to teach for a division, but not told where to report until 10 PM the night before staff re-gathered across the city. On 10 hours notice, I went to the school and began my career. They had no classes, students, or space for me. I slowly carved out a niche that included all three. Until this process was complete, I co-taught with 4 different teachers. In this hectic time, I had no time for preparation or archiving. Reflecting on that experience rehashed a very valuable activity I co-taught with a colleague in Mathematics 9.
I was bursting with excitement to try some new things in math class. I was given my first opportunity on the first day of class. We were to start with the teaching of Order of Operations (or BEDMAS, PEMDAS). The acronyms are convenient, but unfortunately suck all meaning out of the process. I decided to take a different approach, got the students into pairs, and handed out a worksheet. It read as follows:
The 6 key on your calculator is broken. Find the answers to the calculations below. Work out which keys to press before trying it on your calculator. Record the calculations that you do.
Most students were excited to see such a simple activity on day 1. For some reason, students feel safe as long as they are permitted to use a calculator. There is a large debate over the use of calculators in math classrooms. One theory insists that the memorization of rote facts is a necessary skill to be numerate. The other recognizes that the virtual cornucopia of available calculators at any given time diminishes the need for rote memorization. It is more important to understand the number system behind the tables, than the tables themselves. Why put our trust in human memory when it proves to be so fallible? If you don't believe this, spend 30 minutes with my dear grandfather; he will remind you of the deficiencies of human memory. The above tangent probably discloses my personal opinion on the debate. I digress...
Back to the task at hand. The questions begin simple. Students break up the "16" in many different ways--mostly additive. This is a great way to start, because correct answers keep curiosity high. The few students who try breaking up "16" into "8x2" quickly discover that their answer differs from the class. This ripple effect begins discussions.
My circulation fuels the fire. What do you mean you got a different answer? Maybe your substitution was incorrect? After checking that indeed 9+7 was equal to 8x2, we had to re-double our efforts. I prod them onto the next calculations, until I hear that student savior from grade 8...
"Do we have to use BEDMAS?"
Jackpot. I now turn the discussion global. What is BEDMAS, and why do we need it? Cue the history lesson. After that is done, I send them back with this knowledge to work out the nuts and bolts of it. You now have a name, now learn why it works. Students use a hierarchy to solve more difficult problems. They are now careful how the represent the numbers. Maybe "9+7" is different than "8x2". The answers are NEVER worked out with working calculators, I instead collect answers from the class and verify with frequency. This helps divergent answers emerge.
I have the students work left-to-right until the topic of BEDMAS comes up. Then they start using operations for their strength. When students begin to emerge as exceptional, I pose further problems:
What if you weren't given a calculator with brackets, could you write your calculations using operations that would work from left-to-right?
What if I also broke your multiplication button? addition button?
What if I banned a Prime number?
Given certain buttons, which numbers can be created?
Playing with operations and their order brings meaning to the memory cue. Certain metaphors may begin to emerge. Brackets are the toughest operations and "beat up" the weaker ones. Addition is the "little brother" of multiplication. Etc. Keep the experimentation alive!
There are some interesting applets that play with broken calculators. These activities do not necessarily enforce BEDMAS, but mingle it with the knowledge of additive and multiplicative factors. One of my favourites is courtesy of mathisfun.com.
The activity went over very well. I encourage teams of teachers to get together and try exploratory mathematics. The end of the year also reminds me that copious amounts of broken calculators are left for the summer break. The old saying goes that someone's trash is another's treasure. In this case, we can take student waste and turn it into valuable mathematics.